This sounds like a caching issue. A caching plugin like WP-total cache or perhaps a caching system on the server like CloudFlare. I was helping another customer will a similar issue and after exhausting all the options I could think of I asked him to submit a ticket to his web host asking about Cloudflare. I haven’t heard back one way or the other unfortunately. But if more than one person is experiencing this kind of problem it could point to greater adoption of cloudflare by one or several web hosts recently.
Of course the two issues may have separate causes. Please feel free to send over login details for you site via this secure contact form and I will investigate a solution.